条件概率公式 - peter-xbs/CommonCodes GitHub Wiki

条件概率

Conditional independence in general

Conditional independence of two random variables A and B given C holds just in case

(1) p(A, B | C) = p(A | C) * p(B | C)

[cf. R&N's (13.13) on p. 481]. Equivalently, we could have said that A and B are conditionally independent given C just in case B doesn't tell us anything about A if we already know C:

(2) p(A | C) = p(A | B, C)

It's easy to see that (1) and (2) are equivalent by observing that in general

(3) p(A, B | C) = p(A, B, C) / p(C) = p(A | B, C) * p(B, C) / p(C) = p(A | B, C) * p (B | C).

Then it's trivial to derive (1) from (2) and vice versa.